Preheat the oven to 350°F (175°C) and grease a square baking pan.
In a large mixing bowl, combine flour, sugar, cocoa powder, baking powder, baking soda, and salt.
In another bowl, whisk together melted butter, eggs, and vanilla extract.
Combine the wet and dry mixtures, stirring until just incorporated.
Pour the batter into the prepared baking pan and smooth the top.
Bake in the preheated oven for 25-30 minutes, or until the edges are set, but the center remains gooey.
While the cake is baking, prepare the peanut butter topping by mixing peanut butter, powdered sugar, and milk until smooth.
Allow the cake to cool in the pan for 10 minutes before spreading the peanut butter mixture on top.
Spread the peanut butter topping evenly across the warm cake and sprinkle with chocolate chips.
Let the cake cool completely before slicing and serving.